Dynamic programming methods for path-dependent costs:
Hidehiro KaiseOsaka University, Japan 下午 2:30–4:00 鴻經館 107 室Abstract:
Dynamic programming is one of useful methods in control theory and it is used in optimization problems under time-evolving states not only in engineering but also in other wide areas. A conventional dynamic programming treats costs determined by current state but it is recently considered important to work with costs depending on past history of states. In this talk, we overview dynamic programming methods for path- dependent costs in discrete- and continuous-time control problems recently studied in control and stochastic communities. We will see that value functions with path- dependent costs satisfy dynamic programming principles. Under dynamic programming principle, we can expect value functions should satisfy dynamic programming partial differential equations at least formal levels. However we need to specify a notion of a derivative on the state variable because the state space is infinite-dimensional. We introduce and explain several notions on infinite-dimensional calculus which fit to dynamic programming for path-dependent costs: one notion is co-invariant derivative studied in deterministic control community and the other is horizontal and vertical derivatives studied in stochastic community. We will also mention an idempotent expansion method for a particular problem.